Browsing by Subject "Bergman projection"

Now showing 1 - 4 of 4

Open Access
Analytic properties of Besov spaces via Bergman projections
(American Mathematical Society, 2008) Kaptanoğlu, H. T.; Üreyen, A. E.
We consider two-parameter Besov spaces of holomorphic functions on the unit ball of CN: We obtain various exclusions between Besov spaces of di®erent parameters using gap series. We estimate the growth near the boundary and the growth of Taylor coe±cients of functions in these spaces. We ¯nd the unique function with maximum value at each point of the ball in each Besov space. We base our proofs on Bergman projections and imbeddings between Lebesgue classes and Besov spaces. Special cases apply to the Hardy space H2, the Arveson space, the Dirichlet space, and the Bloch space.
Open Access
Carleson measures for Besov spaces on the ball with applications
(Academic Press, 2007) Kaptanoǧlu, Hakkı Turgay
Carleson and vanishing Carleson measures for Besov spaces on the unit ball of CN are characterized in terms of Berezin transforms and Bergman-metric balls. The measures are defined via natural imbeddings of Besov spaces into Lebesgue classes by certain combinations of radial derivatives. Membership in Schatten classes of the imbeddings is considered too. Some Carleson measures are not finite, but the results extend and provide new insight to those known for weighted Bergman spaces. Special cases pertain to Arveson and Dirichlet spaces, and a unified view with the usual Hardy-space Carleson measures is presented by letting the order of the radial derivatives tend to 0. Weak convergence in Besov spaces is also characterized, and weakly 0-convergent families are exhibited. Applications are given to separated sequences, operators of Forelli-Rudin type, gap series, characterizations of weighted Bloch, Lipschitz, and growth spaces, inequalities of Fejér-Riesz and Hardy-Littlewood type, and integration operators of Cesàro type.
Open Access
Harmonic Besov spaces on the ball
(World Scientific Publishing, 2016) Gergün, S.; Kaptanoğlu, H. T.; Üreyen, A. E.
We initiate a detailed study of two-parameter Besov spaces on the unit ball of ℝn consisting of harmonic functions whose sufficiently high-order radial derivatives lie in harmonic Bergman spaces. We compute the reproducing kernels of those Besov spaces that are Hilbert spaces. The kernels are weighted infinite sums of zonal harmonics and natural radial fractional derivatives of the Poisson kernel. Estimates of the growth of kernels lead to characterization of integral transformations on Lebesgue classes. The transformations allow us to conclude that the order of the radial derivative is not a characteristic of a Besov space as long as it is above a certain threshold. Using kernels, we define generalized Bergman projections and characterize those that are bounded from Lebesgue classes onto Besov spaces. The projections provide integral representations for the functions in these spaces and also lead to characterizations of the functions in the spaces using partial derivatives. Several other applications follow from the integral representations such as atomic decomposition, growth at the boundary and of Fourier coefficients, inclusions among them, duality and interpolation relations, and a solution to the Gleason problem. © 2016 World Scientific Publishing Company.
Open Access
Toeplitz operators on arveson and dirichlet spaces
(Birkhaeuser Science, 2007) Alpay, D.; Kaptanoǧlu, H. T.
We define Toeplitz operators on all Dirichlet spaces on the unit ball of CN and develop their basic properties. We characterize bounded, compact, and Schatten-class Toeplitz operators with positive symbols in terms of Carleson measures and Berezin transforms. Our results naturally extend those known for weighted Bergman spaces, a special case applies to the Arveson space, and we recover the classical Hardy-space Toeplitz operators in a limiting case; thus we unify the theory of Toeplitz operators on all these spaces. We apply our operators to a characterization of bounded, compact, and Schatten-class weighted composition operators on weighted Bergman spaces of the ball. We lastly investigate some connections between Toeplitz and shift operators. © Birkhäuser Verlag Basel/Switzerland 2007.